I don't use hair much, but with some experimentation I found the following solution. Perhaps someone with more experience has a better approach, but this is what I came up with.
The basic hair material does not include luminance or glow channels, so to get a neon look for hair select one of the polygon modes in the Generate menu (circle for example) which will then allow you to add a normal material to the hair object. Add luminance (and glow if desired) to that material, and control the hair properties (thickness, frizz, curl, etc.) in the hair material. In my example below I added a light with its intensity at 0% in order to disable the default light and thus eliminate all light from the scene. Using a polygon mode in the Generate menu generates hair as polygons, so it will increase render times and memory requirements substantially (see Hair Limitations in the help menu). Depending upon your project, and your hardware, this may or may not be a suitable solution.
That's just brilliant. Thank you for taking the time to look into this. I've just had a look at your file and transfered your settings to mine and it works like a dream. It will, as you obviously realise, increase the render time but it is a solution! Later on in the summer I'm planning to get some souped-up kit, which will make lengthy animations of this sort far more practical. Assuming I get that far, I'll send you the result.